A change in path length can have a significant effect on the interference pattern observed when light passes through two slits, known as the double-slit interference pattern. The interference pattern is created by the constructive and destructive interference of light waves from the two slits.
When light passes through the two slits, it forms two separate wavefronts that spread out and overlap. If the path lengths from the slits to a particular point on the screen are equal, the waves arrive in phase and reinforce each other, creating a bright region on the screen. This is known as constructive interference.
However, if there is a difference in the path lengths, the waves arriving at a particular point on the screen can be out of phase with each other. In this case, the waves interfere destructively, leading to a dark region on the screen.
The interference pattern is determined by the phase difference between the waves arriving at a given point. The phase difference depends on the wavelength of light, the distance between the slits (known as the slit separation or slit spacing), and the path length difference between the two waves.
When there is a change in the path length, such as by introducing a material of different refractive index in one of the paths or by altering the physical length of one of the paths, the phase difference between the waves at a particular point on the screen can change.
If the path length difference changes by an integer multiple of the wavelength (λ) of the light, the interference pattern remains unaffected because the phase difference remains the same. This corresponds to a whole number of complete wave cycles.
However, if the path length difference changes by a fraction of the wavelength (λ), the interference pattern will be modified. This is because the phase difference between the waves will change, leading to a shift in the constructive and destructive interference regions on the screen.
For example, if the path length difference increases by λ/2, the dark fringes can shift to positions where the bright fringes were originally located, and vice versa. This is known as a half-wavelength phase shift and causes the interference pattern to shift.
In summary, a change in path length affects the interference pattern by altering the phase difference between the waves arriving at a particular point on the screen. This can result in a shift or modification of the interference fringes observed in the double-slit experiment.